I have a dell inspiron mini 10 (1010) and last night i was installing a game on it i think it was World of Warcraft then later on that night i saw the white power light cut off, So this morning i tried plugging it into the AC adapter and then the white power light comes on then it goes to the dell boot screen then shuts right back off. I did order a replacement battery and it came yesterday afternoon and it was working perfectly before the night i was installing World of Warcraft on it.
Welcome to the Community. Remove the battery, and the ac adapter, press and hold and try power button for 15 seconds, and then try powering on with ac adapter alone.
If it lets you run diagnostics, press the F12 key at the Dell Logo on startup, select diagnostics and run an extended diagnostics on your computer.